1 Integers, Powers And Roots - Cambridge University Press
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1 Integers, powers and rootsThe first primes are 2 3 5 7 11 13 17 19 23 29 . . .Prime numbers have just two factors: 1 and the number itself.Every whole number that is not prime can be written as a productof prime numbers in exactly one way (apart from the order ofthe primes).8 2 2 2 65 5 132527 7 19 19132 2 2 3 11It is easy to multiply two prime numbers. For example,13 113 1469.It is much harder to do the inverse operation. For example,2021 is the product of two prime numbers. Can you find them?This fact is the basis of a system that is used to encode messagessent across the internet.The RSA cryptosystem was invented by Ronald Rivest,Adi Shamir and Leonard Adleman in 1977. It uses twolarge prime numbers with about 150 digits each. Theseare kept secret. Their product, N, with about 300 digits, is madepublic so that anyone can use it.Key wordsMake sure you learn andunderstand these key words:integerinversemultiplecommon multiplelowest common multiple (LCM)factorcommon factorhighest common factor (HCF)prime numberprimefactor treepowerindex (indices)squarecubesquare rootcube rootIf you send a credit card number to awebsite, your computer performs acalculation with N and your credit cardnumber to encode it. The computerreceiving the coded number will doanother calculation to decode it. Anyoneelse, who does not know the factors, willnot be able to do this.Prime numbers more than 200 are 211 223 227 229 233 239 241 251 257 263 269 271 . . . . . .1 in this web service Cambridge University PressIntegers, powers and roots7www.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1.1 Arithmetic with integers1.1 Arithmetic with integersIntegers are whole numbers. They may be positive or negative. Zero is also an integer.You can show integers on a number line.–5–4–3–2–101232 3 52 2 42 1 342 0 25Look at the additions in the box to the right. The number added to 2 decreases, or goesdown, by 1 each time. The answer also decreases, or goes down, by 1 each time.2 1 12 2 02 3 12 4 2Now see what happens if you subtract. Look at the first column.The number subtracted from 5 goes down by 1 each time. The answergoes up by 1 each time. Now look at the two columns together.You can change a subtraction into an addition by adding the inversenumber. The inverse of 3 is 3. The inverse of 3 is 3.For example, 5 – –3 5 3 8.5 3 25 3 25 2 35 2 35 1 45 1 45 0 55 0 55 1 65 1 65 2 75 2 75 3 85 3 8Worked example 1.1aWork these out.a 3 7a 3 7 4b 5 8 13c 3 9 6Subtract 7 from 3.The inverse of 8 is 8.The inverse of 9 is 9.Look at these multiplications.b 5 83 5 15c 3 93 7 4 5 8 5 8 13 3 9 3 9 6The pattern continues like this. 1 5 52 5 10 2 5 101 5 5 3 5 150 5 0 4 5 20You can see that negative integer positive integer negative answer.Now look at this pattern.The pattern continues like this. 3 4 12 3 1 3 3 3 9 3 2 6 3 2 6 3 3 9 3 1 3 3 4 12 3 0 0 3 5 15You can see that negative integer negative integer positive answer.81Integers, powers and roots in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1.1 Arithmetic with integersHere is a simple rule, which also works for division.When you multiply two integers:if they have same signs positive answerif they have different signs negative answerWorked example 1.1bWork these out.a 12 3b 8 5abcd12 3 368 5 4020 4 524 6 4TheTheTheThe12 3 36 8 5 40 20 4 5 24 6 4signssignssignssignsarearearearec 20 4d 24 6different so the answer is negative.the same so the answer is positive.different so the answer is negative.the same so the answer is positive.Warning: This rule works for multiplication and division. It does not work for addition or subtraction. Exercise 1.11 Work out these additions.a 3 6b 3 8c 10 4d 10 7e 12 42 Work out these additions.a 30 20b 100 80c 20 5d 30 70e 45 403 Work out these subtractions.a 4 6b 4 6c 6 4d 6 6e 2 104 Write down additions that have the same answers as these subtractions. Then work out the answerto each one.a 4 6b 4 6c 8 2d 4 6e 12 105 Work out these subtractions.a 7 2b 5 3c 12 4d 6 6e 2 106 Here are some addition pyramids. Each number is the sum of the two inthe row below it.Copy the pyramids. Fill in the missing numbers.bcdaIn part a, 3 5 2e3 333 3–2–2–2–2–2–7–7–7–7–72 222 2–6–6–6–6–63 3 3 3–53–5–5–51–51 1 1 1 –2–2–2–2–3–2–3–3–35–35 5 5 5 2 2 2 –3–3–32 222 27 Here is a subtraction table. Two answers have already been filled in: 4 4 8 and 4 2 6.Copy the table and complete it.firstnumber 420 2 4 48second number 2024 61 in this web service Cambridge University PressIntegers, powers and roots9www.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1.1 Arithmetic with integers8 Work out these multiplications.a 5 4b 8 6c 4 5d 6 10e 2 209 Work out these divisions.a 20 10b 30 6c 12 4d 50 5e 16 4d 40 8e 12 410 Write down two correct division expressions.a 4 10b 20 5c 20 511 Here are some multiplications. In each case, use the same numbers to write downtwo correct division expressions.a 5 3 15b 8 4 32c 6 7 4212 Here is a multiplication table. Three answers have already been filled in. 3210 1 2 3abcd 3 2 101263 23Copy the table and complete it.Colour all the 0 answers in one colour, for example, green.Colour all the positive answers in a second colour, for example, blue.Colour all the negative answers in a third colour, for example, red.The product is the result ofmultiplying two numbersIn part a, 2 3 613 These are multiplication pyramids. Each number is the product of thetwo in the row below it.Copy each pyramid. Fill in the missing –12–122 2–4–4–3–35 5–2–2–1–1 –4–4–3–35 5–3–3–1–13 �123 3–3–3646422–16–162 214 a What integers will replace the symbols to make this multiplication correct? 12b How many different pairs of numbers can you find that give this answer?15 Work these out.a 5 3b 5 3c 4 516 Write down the missing numbers.b 2 6a 4 20f 4 3e 2 2101d 60 10c 5 2e 2 18df 10 4 3 12Integers, powers and roots in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1.2 Multiples, factors and primes1.2 Multiples, factors and primes6 1 66 2 12 6 3 18 9 1 9 9 2 18 9 3 27 The multiples of 6 are 6, 12, 18, 24, 30, 36, , The multiples of 9 are 9, 18, 27, 36, 45, 54, , The common multiples of 6 and 9 are 18, 36, 54, 72, , 18 36 54 are in both lists of multiples.The lowest common multiple (LCM) of 6 and 9 is 18.The factors of a number divide into it without a remainder.3 6 18 so 3 and 6 are factors of 18The factors of 18 are 1, 2, 3, 6, 9 and 18.The factors of 27 are 1, 3, 9 and 27.The common factors of 18 and 27 are 1, 3 and 9.The highest common factor (HCF) of 18 and 27 is 9.Some numbers have just two factors. Examples are 7 (1 and 7 are factors), 13 (1 and 13 are factors)and 43. Numbers with just two factors are called prime numbers or just primes. The first ten primesare 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.Worked example 1.2aaFind the factors of 45.b Find the prime factors of 48.aThe factors of 45 are 1, 3, 5, 9, 15and 45.45 1 45 so 1 and 45 are factors. (1 is always a factor.)Check 2, 3, 4, in turn to see if it is a factor.2 is not a factor. (45 is an odd number.)45 3 153 and 15 are factors.4 is not a factor.45 5 95 and 9 are factors.6, 7 and 8 are not factors. The next number to try is 9 but wealready have 9 in the list of factors. You can stop when youreach a number that is already in the list.bThe prime factors of 48 are 2 and 3.You only need to check prime numbers.48 2 242 is a prime factor. 24 is not.48 3 163 is a prime factor. 16 is not.5 and 7 are not factors.Because 7 7 is bigger than 48, you can stop there.Worked example 1.2bFind the LCM and HCF of 12 and 15.The LCM is 60.The multiples of 12 are 12, 24, 36, 48, 60, , .The multiples of 15 are 15, 30, 45, 60, 75, , 60 is the first number that is in both lists.The HCF is 3.The factors of 12 are 1, 2, 3, 4, 6 and 12.The factors of 15 are 1, 3, 5 and 15.3 is the largest number that is in both lists.1 in this web service Cambridge University PressIntegers, powers and roots11www.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1.2 Multiples, factors and primes Exercise 1.21 Find the factors of each number.a 20b 27c 75d 232 Find the first four multiples of each number.a 8b 15c 7d 20e 100f 98e 33f 1003 Find the lowest common multiple of each pair of numbers.a 6 and 8b 9 and 12c 4 and 14d 20 and 30 e 8 and 32f 7 and 114 The LCM of two numbers is 40. One of the numbers is 5. What is the other number?5 Find:a the factors of 24b the factors of 32d the highest common factor of 24 and 32.c the common factors of 24 and 326 List the common factors of each pair of numbers.a 20 and 25b 12 and 18c 28 and 35d 8 and 24e 21 and 32f 19 and 317 Find the HCF of the numbers in each pair.a 8 and 10b 18 and 24c 40 and 50d 80 and 100e 17 and 33f 15 and 308 The HCF of two numbers is 8. One of the numbers is between 20 and 30. The other number isbetween 40 and 60. What are the two numbers?9 31 is a prime number. What is the next prime after 31?10 List the prime numbers between 60 and 70.11 Read what Xavier and Alicia say about the number 91.91 is a prime number.91 is not a prime number.Who is correct? Give a reason for your answer.12 73 and 89 are prime numbers. What is their highest common factor?13 7 is a prime number. No multiple of 7, except 7 itself, can be a prime number. Explain why not.14 List the prime factors of each number.a 12b 15c 21d 49e 30f 7715 a Write down three numbers whose only prime factor is 2.b Write down three numbers whose only prime factor is 3.c Write down three numbers whose only prime factor is 5.16 Find a number bigger than 10 that has an odd number of factors.17 Find a number that has three prime factors.121Integers, powers and roots in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1.3 More about prime numbers1.3 More about prime numbersAny integer bigger than 1, that is not prime, can be written as a product of prime numbers.Here are some examples.84 2 2 3 745 3 3 5196 2 2 7 7You can use a factor tree to find and show factors.This is how to draw a factor tree for 120.1201 Draw branches to two numbers that multiply to make 120. Here 12 and10 are chosen.12102 Do the same with 12 and 10. 12 3 4 and 10 2 53 3, 2 and 5 are prime, so stop.34 24 4 2 2 so draw two branches.5 Stop, because all the end numbers are prime.226 Multiply all the numbers at the ends of the branches.120 2 2 2 3 5You can draw the tree in different ways.Here is a different tree for 120.The numbers at the ends of the branches are the same.You can write the result like this.120 23 3 5.The small number ³ next to the 2 is called an index. 23 means 2 2 2.1202602305Check that these are correct.60 2² 3 575 3 5²You can use these expressions to find the LCM and HCF of 60 and 75.For the LCM, take the larger frequency of each prime factor and multiplythem all together.LCM 2² 3 5² 4 3 25 300For the HCF, take the smaller frequency of each prime factor that occurs inboth numbers and multiply them all together.HCF 3 5 1562360 22 3 575 3 52Two 2s, one 3, two 5s60 22 3 575 3 52No 2s, one 3, one 51 in this web service Cambridge University Press5Integers, powers and roots13www.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1.3 More about prime numbers Exercise 1.31 Copy and complete each of these factor trees.ba481006810c101082542 a Draw a different factor tree for each of the numbers in question 1.b Write each of these numbers as a product of primes.i 48ii 100iii 1083 Match each number to a product of primes.One has been done for you.20244250180 22 52 3 722 32 52 5223 34 Write down the number that is being represented.b 2 33c 3 112d 23 72a 22 3 55 Write each number as a product of primes.a 24b 50c 72d 200e 165f 136i 45ii 757 a Write each number as a product of primes.b Find the LCM of 90 and 140.c Find the HCF of 90 and 140.i 90ii 1401f 52 13You can use a factor tree to help you.6 a Write each number as a product of primes.b Find the LCM of 45 and 75.c Find the HCF of 45 and 75.8 37 and 47 are prime numbers.a What is the HCF of 37 and 47?14e 24 3²b What is the LCM of 37 and 47?Integers, powers and roots in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1.4 Powers and roots1.4 Powers and rootsA number multiplied by itself several times is called a power ofThe plural of index is indices:that number. You use indices to show powers.one index, two indices.Here are some powers of 5.5² 5 5 25This is five squared or the square of five.35 5 5 5 125This is five cubed or the cube of five.54 5 5 5 5 625This is five to the power four.55 5 5 5 5 5 3125 This is five to the power five.The square of 5 is 5² 25.Therefore the square root of 25 is 5 and you write this as 25 5.The cube of 5 is 53 125.Therefore the cube root of 125 is 5 and you write this as 3 125 5.means square root.3means cube root.5 is not the only square root of 25.( 5)2 5 5 25 so 25 has two square roots, 5 and 5.25 means the positive square root.125 only has one integer cube root. 5 is not a cube root because 5 5 5 125.Square numbers have square roots that are integers.Examples: 132 169 so 169 13 192 361 so 361 19Try to memorise these five cubes and their corresponding cube roots:13 1 so 3 1 123 8 so 3 8 233 27 so 3 27 343 64 so 3 64 4 53 125 so 3 125 5 Exercise 1.41 Find the value of each power.a 3²b 33c 342 Find the value of each power.a 10²b 10³c 104d 35Over one billionpeople live in India.3 106 is one million and 109 is one billion.Write down these two numbers in full.4 In each pair, which of the two numbers is larger?b 26 or 62c 54 or 45a 35 or 535 a N 3 is 27. What number is N ?b 6M is 1296. What number is M ?1 in this web service Cambridge University PressIntegers, powers and roots15www.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore information1.4 Powers and roots6 Can you find two different integers, A and B, so that AB BA?7 Write down two square roots for each of these numbers.a 9b 36c 81d 1968 Read what Maha says abouther number. What could hernumber be?e 225f 400I am thinking of a number. It is between 250 and350. Its square root is an integer.9 Read what Hassan says about hisnumber. What is the largest possiblevalue of his number?I am thinking of a number. It is less than 500.Its cube root is an integer.10 Find the value of each of these.a100b4003c27d11 Read what Oditi says about her number.Find a possible value for her number.12 210 1024. Use this fact to find:b 212a 2113125e31000I am thinking of a number. Its square rootis an integer. Its cube root is an integer.c 2913 a Find the value of each expression. i 1³ 2³b Find the value of 13 23 33 .ii13 23c Find the value of 13 23 33 4 3 .d Can you see an easy way to work out the value of 13 23 33 43 53 ? If so, describe it.SummaryYou should now know that:You should be able to: You can multiply or divide two integers. If theyhave the same sign the answer is positive( 5 2 10). If they have different signs theanswer is negative ( 5 2 10). Add, subtract, multiply and divide integers. You can subtract a negative number by adding thecorresponding positive number. You can find multiples of a number by multiplyingby 1, 2, 3, etc. Prime numbers have just two factors. You can write every positive integer as a productof prime factors. You can use the products of prime factors to findthe lowest common factor and highest commonmultiple. 54 means 5 5 5 5. Positive integers have two square roots. 49 7 and 3 64 4.161 Identify and use multiples and factors. Identify and use primes. Find common factors and highest common factors(HCF). Find lowest common multiples (LCM). Write a number in terms of its prime factors, forexample, 500 22 53. Calculate squares, positive and negative squareroots, cubes and cube roots. Use index notation for positive integer powers. Calculate accurately, choosing operations andmental or written methods appropriate to thenumbers and context. Manipulate numbers and apply routinealgorithms, for example, to find the HCF and LCMof two numbers.Integers, powers and roots in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-1-107-69787-4 – Cambridge Checkpoint MathematicsGreg Byrd Lynn Byrd and Chris PearceExcerptMore informationEnd-of-unit reviewEnd-of-unit review1 Work these out.a 5 3b 3 5c 8 7d 3 13e 7 72 Work these out.a 2 5b 3 4c 12 5d 5 12e 9 93 Work these out.a 3 9b 8 4c 20 4d 30 5e 16 84 Copy and complete this multiplication table. 235 4 36305 Here is a number chain. Each number is the product of the previous two numbers. 22 4 1Write down the next two numbers in the chain.6 Find all the factors of each number.a 42b 52c 55d 29e 64f 697 a Find two prime numbers that add up to 40.b Find another two prime numbers that add up to 40.c Are there any more pairs of prime numbers that add up to 40? If so, what are they?8 Write each of these numbers as a product of its prime factors.a 18b 96c 200d 240e 135f 1759 Use your answers to question 8 to find:a the highest common factor of 200 and 240c the lowest common multiple of 18 and 96b the highest common factor of 135 and 175d the lowest common multiple of 200 and 240.10 Find the square roots of each number.a 25b 81c 169d 25611 Find the value of each number.a64b36412 In computing, 210 is called 1K. Write down as a number:a 1Kb 2Kc 4K.13 a Read Shen’s comment. What mistake has he made?b Correct the statement.35 and 53 are both equal to 15.14 The HCF of two numbers is 6. The LCM is 72. One of the numbers is 24.Find a possible value of the other number.1 in this web service Cambridge University PressIntegers, powers and roots17www.cambridge.org
Cambridge University Press 978-1-107-69787-4 – Cambridge Checkpoint Mathematics Greg Byrd Lynn Byrd and Chris Pearce Excerpt More information 1 Integers, powers and roots 1 8 Find: 9 10 11 12
2 Integers and Introduction to Solving Equations 98 2.1 Introduction to Integers 99 2.2 Adding Integers 108 2.3 Subtracting Integers 116 2.4 Multiplying and Dividing Integers 124 Integrated Review—Integers 133 2.5 Order of Operations 135 2.6 Solving Equations: The Addition and Multiplication Properties 142 Group Activity 151 ocabulary CheckV 152
Write general rules for adding (a) two integers with the same sign, (b) two integers with different signs, and (c) an integer and its opposite. Use what you learned about adding integers to complete Exercises 8 –15 on page 12. 1.2 Lesson 10 Chapter 1 Operations with Integers Adding Integers with the Same Sign
of Ten Using Base Ten Blocks Powers of Powers . Powers of Powers Multiplying By Powers of Ten Powers of Powers Use these activities to help your students develop knowledge of place value and powers of 10 to support multiplicative thinking . Comparing Decimals Arrow Cards Compare Decimals using
Roots of complex numbers Every number has two square roots. The square roots of 16 are: The square roots of 24 are: The square roots of -81 are: The square roots of -75 are: Likewise, every number has three cube roots, four fourth roots, etc. (over the complex number system.) So if we want to find the four fo
students have mastered addition of integers, subtraction of integers may be introduced. 4 Subtraction of Integers In a subtraction equation, a b c,a is called the minuend ,b is the subtrahend , and c is the difference . Example 5 involves subtracting integers with same signs and Example 6 involves subtracting integers
Substituting Variable Values in Program Understand how real numbers and integers are treated by the CNC control Real numbers are any number rational or irrational Real numbers include integers 1.25, -.3765, 5, -10 are all real numbers Integers are whole numbers 2500, 3, 20 Numbers with decimals are not integers i.e. 1.25 Some NC words only allow integers
POWERS OF ATTORNEY ACT 2003: A COMMENTARY 6 POWERS OF ATTORNEY ACT 2003: COMMENTARY The commentary is provided in black text. Reference to the "Act" is a reference to the Powers of Attorney Act 2003 as amended. Reference to the "Regulation" is a reference to the Powers of Attorney Regulation 2011, recently amended by the Powers of Attorney Amendment Act 2013 and the Powers of
Academic Writing Certain requirements pertain to work written by students for higher education programmes. If you are a new student or perhaps returning to study after a break you may feel that you need help with developing appropriate skills for academic writing. This section is designed to help you to meet the requirements of the School in relation to academic writing. Continuous assessment .
